Finite Group Databasebeta

← #11781498

Subgroups — #11781498

Order 1152.

374 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#4830C211yesmaximal, characteristic, complemented
2#10590384C211yesmaximal, characteristic, complemented
3#10590367C211yesmaximal, characteristic
4#10590281C211yesmaximal, characteristic
5#10589351C211yesmaximal, characteristic
6C3×#4486C211yesmaximal, characteristic, complemented
7#10589099C211yesmaximal, characteristic, complemented
8#70941—31nomaximal
9#68470—31nomaximal
10#64083C2211yescharacteristic
11C3×#869C2211yescharacteristic
12C3×C4×Dic6C2211yescharacteristic
13C2×#3718—21no—
14C2×#3717—21no—
15C32×C4⋊C8C2211yescharacteristic
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×Dic3⋊C8—21no—
18C3×C8×Dic3—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4×C3⋊C8C2211yescharacteristic
21#63565—21no—
22#63564—21no—
23#63564—21no—
24#63564—21no—
25C3×C8×Q8—31no—
26#5090—31no—
27#5088—31no—
28#5031—31no—
29#5027—31no—
30#4848—31no—
31#4830D311yescharacteristic, complemented
32#4694—31no—
33C3×#332—31no—
34#4486D311yescharacteristic, complemented
35#4485—31no—
36#4485—31no—
37#4485—61no—
38#4479—31no—
39#4458—31no—
40C2×C4⋊(C3⋊D3)—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C4⋊(C3⋊D3)—21no—
43C2×C3×Dic6—21no—
44C2×C3×Dic6D411yescharacteristic
45C2×C32×C8—21no—
46C32×C42C2311yescharacteristic
47C2×C3⋊(C3⋊C8)—21no—
48C3×C4⋊Dic3—21no—
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3D411yescharacteristic
51C3×Dic3⋊C4—21no—
52C3×C4×Dic3—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#3718—42no—
57#3717—42no—
58#1682—91noSylow
59C3×C4⋊Q8—31no—
60C3×C4×Q8—31no—
61C2×#216—61no—
62C2×#216—61no—
63C2×#215—61no—
64C2×#204—61no—
65#874—31no—
66#874—31no—
67#874—61no—
68#869C2×D311yescharacteristic
69C4×Dic6C2×D311yescharacteristic
70C3×C4⋊C8—21no—
71C3×C4⋊C8—31no—
72C3×C4⋊C8—61no—
73C3×C4⋊C8C2×D311yescharacteristic
74C3×C4⋊C8C2×D311yescharacteristic
75C3×C4×C8—31no—
76C3×C4×C8—61no—
77#835—61no—
78#821—61no—
79#821—61no—
80#821—61no—
81Dic3⋊C8—21no—
82C8×Dic3—21no—
83#810—61no—
84#809—61no—
85#809—61no—
86#809—61no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4×C3⋊C8C2×D311yescharacteristic
92C22×C3⋊D3—41no—
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4#4111yescharacteristic
96C4⋊(C3⋊D3)—41no—
97C4⋊(C3⋊D3)—41no—
98C4⋊(C3⋊D3)—41no—
99C4⋊(C3⋊D3)—41no—
100C4⋊(C3⋊D3)—81no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—41no—
104C3×Dic6—21no—
105C3×Dic6—41no—
106C3×Dic6—41no—
107C3×Dic6D812yes—
108C32×C8—41no—
109C3⋊(C3⋊C8)—41no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—21no—
112C3×C3⋊C8—41no—
113#486—91no—
114#462—91no—
115#460—91no—
116C8×Q8—31no—
117#335—91no—
118#332—31no—
119#331—91no—
120C2×C3×Q8—31no—
121C2×C3×Q8—61no—
122C2×D12—61no—
123C2×D12—61no—
124C2×D12—61no—
125C2×D12—61no—
126C2×D12—62no—
127C2×D12—61no—
128C2×D12—61no—
129C2×D12—121no—
130C2×D12—121no—
131C2×Dic6—21no—
132C2×Dic6D1211yescharacteristic
133C2×C3×C8—21no—
134C2×C3×C8—21no—
135C2×C3×C8—31no—
136C2×C3×C8—31no—
137C2×C3×C8—41no—
138C2×C3×C8—61no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—61no—
142C3×C4⋊C4—61no—
143C3×C42—21no—
144C3×C42—61no—
145C3×C42C22×D311yescharacteristic
146C3×C42C22×D311yescharacteristic
147#216—122no—
148#216—122no—
149#215—122no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3D1211yescharacteristic
153Dic3⋊C4—21no—
154C4×Dic3—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—61no—
157C2×C3⋊C8—61no—
158C2×C3⋊C8—121no—
159C2×C3⋊C8#6711yescharacteristic
160C2×C3⋊C8#6711yescharacteristic
161#204—122no—
162C22×C32#12111yescharacteristic
163C2×C3⋊D3—81no—
164C2×C3⋊D3—81no—
165C2×C3⋊D3—81no—
166C32×C4—21no—
167C32×C4C2×D811yescharacteristic
168C32×C4#13511yescharacteristic
169C32×C4#13511yescharacteristic
170C32×C4#13711yescharacteristic
171C3×Dic3—22no—
172C3×Dic3—41no—
173C3×Dic3—81no—
174C2×Q16—181no—
175C2×SD16—181no—
176C4⋊Q8—31no—
177#127—91no—
178C4×Q8—31no—
179C4⋊C8—31no—
180C4⋊C8—61no—
181C4⋊C8—91no—
182C4⋊C8D3211yescharacteristic
183#103—181no—
184#102—181no—
185#102—181no—
186#102—181no—
187C4×C8—31no—
188C4×C8—61no—
189C22×D3—122no—
190C22×D3—121no—
191C22×D3—121no—
192C3×Q8—32no—
193C3×Q8—61no—
194C3×Q8—121no—
195C3×Q8—121no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—121no—
202C2×C3×C4C2×D1211yescharacteristic
203C2×C3×C4#23511yescharacteristic
204C2×C3×C4#23511yescharacteristic
205C2×C3×C4D3×D411yescharacteristic
206C2×C3×C4D3×D411yescharacteristic
207C2×C3×C4C2×#6711yescharacteristic
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—41no—
211D12—121no—
212D12—121no—
213D12—121no—
214D12—121no—
215D12—121no—
216D12—121no—
217D12—121no—
218D12—121no—
219D12—241no—
220D12—241no—
221D12—241no—
222D12—241no—
223D12—241no—
224D12—241no—
225D12—242no—
226Dic6—21no—
227Dic6—41no—
228Dic6—41no—
229Dic6D2412yes—
230C3×C8—41no—
231C3×C8—41no—
232C3×C8—61no—
233C3×C8—61no—
234C3×C8—81no—
235C3×C8—121no—
236C3⋊C8—21no—
237C3⋊C8—21no—
238C3⋊C8—41no—
239C3⋊C8—121no—
240C3⋊C8—121no—
241C3⋊C8—241no—
242C2×C32#46411yescharacteristic
243C2×C32#46611yescharacteristic
244C2×C32#47111yescharacteristic
245C3⋊D3—161no—
246C2×Q8—31no—
247C2×Q8—61no—
248C2×D4—181no—
249C2×D4—181no—
250C2×D4—181no—
251Q16—362no—
252SD16—362no—
253C2×C8—21no—
254C2×C8—31no—
255C2×C8—31no—
256C2×C8—61no—
257C2×C8—181no—
258C4⋊C4—31no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C42—61no—
263C42C2×D3211yescharacteristic
264C22×C3—21no—
265C22×C3#88411yescharacteristic
266C22×C3#93011yescharacteristic
267C2×D3—242no—
268C2×D3—242no—
269C2×D3—242no—
270C2×D3—241no—
271C2×D3—241no—
272C2×D3—241no—
273C2×D3—241no—
274C2×D3—241no—
275C2×D3—241no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—41no—
283C3×C4—62no—
284C3×C4—121no—
285C3×C4—241no—
286C3×C4C2×D2411yescharacteristic
287C3×C4#90411yescharacteristic
288C3×C4D3×D811yescharacteristic
289C3×C4#91611yescharacteristic
290C3×C4#91611yescharacteristic
291C3×C4#91811yescharacteristic
292C3×C4#94211yescharacteristic
293C3×C4#95011yescharacteristic
294Dic3—22no—
295Dic3—41no—
296Dic3—81no—
297C32#168211yescharacteristic, complemented, Sylow
298C23—361no—
299Q8—32no—
300Q8—61no—
301Q8—121no—
302Q8—121no—
303D4—361no—
304D4—361no—
305D4—361no—
306D4—361no—
307D4—721no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—121no—
311C2×C4#379811yescharacteristic
312C2×C4D3×D1211yescharacteristic
313C2×C4D3×#6711yescharacteristic
314C8—41no—
315C8—61no—
316C8—61no—
317C8—121no—
318C8—361no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3#475411yescharacteristic
323C2×C3#477511yescharacteristic
324C2×C3#479811yescharacteristic
325C2×C3#502811yescharacteristic
326C2×C3#511511yescharacteristic
327C2×C3#514111yescharacteristic
328D3—482no—
329D3—481no—
330D3—481no—
331C22—721no—
332C22—721no—
333C22—721no—
334C22#6390611yescharacteristic
335C4—21no—
336C4—62no—
337C4—121no—
338C4—241no—
339C4D3×D2411yescharacteristic
340C4#6380611yescharacteristic
341C4#6393511yescharacteristic
342C4#6394811yescharacteristic
343C3—21no—
344C3#6847011yescharacteristic, complemented
345C3#7094111yescharacteristic, complemented
346C2—1441no—
347C2#1059053611yescharacteristic
348C2#1059082611yescharacteristic
349C2#1059099811yescharacteristic
subgroupG/H
centerC22#63906
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#121
prime radical#64083C22
hypercenterC4⋊C8D32
hypocenterC32#1682
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#4830C2semidirect
2. #10590384C2semidirect
6. C3×#4486C2semidirect
7. #10589099C2semidirect
31. #4830D3semidirect
34. #4486D3semidirect
297. C32#1682semidirect
344. C3#68470semidirect
345. C3#70941semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#48302
#105903842
#105903672
#105902812
#105893512
C3×#44862
#105890992
#709413
#684703